Collapsing manifolds obtained by Kummer-type constructions

Paternain, Gabriel; Petean, Jimmy

doi:10.1090/S0002-9947-09-04704-7

Collapsing manifolds obtained by Kummer-type constructions
HTML articles powered by AMS MathViewer

by Gabriel P. Paternain and Jimmy Petean PDF

Trans. Amer. Math. Soc. 361 (2009), 4077-4090 Request permission

Abstract:

We construct $\mathcal {F}$-structures on a Bott manifold and on some other manifolds obtained by Kummer-type constructions. We also prove that if $M=E\# X$, where $E$ is a fiber bundle with structure group $G$ and a fiber admitting a $G$-invariant metric of non-negative sectional curvature and $X$ admits an $\mathcal {F}$-structure with one trivial covering, then one can construct on $M$ a sequence of metrics with sectional curvature uniformly bounded from below and volume tending to zero (i.e. $\operatorname {Vol}_K(M)=0$). As a corollary we prove that all the elements in the Spin cobordism ring can be represented by manifolds $M$ with $\operatorname {Vol}_K (M)=0$.

References

R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 251664, DOI 10.1090/S0002-9947-1969-0251664-4
Jeff Cheeger and Mikhael Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. I, J. Differential Geom. 23 (1986), no. 3, 309–346. MR 852159
Jeff Cheeger and Mikhael Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. II, J. Differential Geom. 32 (1990), no. 1, 269–298. MR 1064875
Kenji Fukaya and Takao Yamaguchi, The fundamental groups of almost non-negatively curved manifolds, Ann. of Math. (2) 136 (1992), no. 2, 253–333. MR 1185120, DOI 10.2307/2946606
Michael Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99 (1983). MR 686042
Mikhael Gromov and H. Blaine Lawson Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), no. 3, 423–434. MR 577131, DOI 10.2307/1971103
Dominic D. Joyce, Compact Riemannian $7$-manifolds with holonomy $G_2$. I, II, J. Differential Geom. 43 (1996), no. 2, 291–328, 329–375. MR 1424428
Dominic D. Joyce, Compact Riemannian $7$-manifolds with holonomy $G_2$. I, II, J. Differential Geom. 43 (1996), no. 2, 291–328, 329–375. MR 1424428
D. D. Joyce, Compact $8$-manifolds with holonomy $\textrm {Spin}(7)$, Invent. Math. 123 (1996), no. 3, 507–552. MR 1383960, DOI 10.1007/s002220050039
Claude LeBrun, Four-manifolds without Einstein metrics, Math. Res. Lett. 3 (1996), no. 2, 133–147. MR 1386835, DOI 10.4310/MRL.1996.v3.n2.a1
Claude LeBrun, Ricci curvature, minimal volumes, and Seiberg-Witten theory, Invent. Math. 145 (2001), no. 2, 279–316. MR 1872548, DOI 10.1007/s002220100148
John Lott, $\hat A$-genus and collapsing, J. Geom. Anal. 10 (2000), no. 3, 529–543. MR 1794576, DOI 10.1007/BF02921948
Gabriel P. Paternain and Jimmy Petean, Minimal entropy and collapsing with curvature bounded from below, Invent. Math. 151 (2003), no. 2, 415–450. MR 1953264, DOI 10.1007/s00222-002-0262-7
Jimmy Petean, The Yamabe invariant of simply connected manifolds, J. Reine Angew. Math. 523 (2000), 225–231. MR 1762961, DOI 10.1515/crll.2000.049
Takashi Shioya and Takao Yamaguchi, Collapsing three-manifolds under a lower curvature bound, J. Differential Geom. 56 (2000), no. 1, 1–66. MR 1863020
Takashi Shioya and Takao Yamaguchi, Volume collapsed three-manifolds with a lower curvature bound, Math. Ann. 333 (2005), no. 1, 131–155. MR 2169831, DOI 10.1007/s00208-005-0667-x
Stephan Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 136 (1992), no. 3, 511–540. MR 1189863, DOI 10.2307/2946598
Chanyoung Sung, Surgery, curvature, and minimal volume, Ann. Global Anal. Geom. 26 (2004), no. 3, 209–229. MR 2097617, DOI 10.1023/B:AGAG.0000042898.55061.ef
C.Z. Tan, Ph.D. thesis, University of Cambridge, 2006.
C.T.C. Wall, Classification problems in differential topology, V. On certain $6$-manifolds, Invent. Math. 1 (1966) 355–374.
Takao Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. (2) 133 (1991), no. 2, 317–357. MR 1097241, DOI 10.2307/2944340